26 INVESTIGATION of fome 
Now-Ac + Aa =r-+cO +7r—cO =ar4 2x cO 
Shae = pO AOR ay 1 LO 
Cd +b = Pa Od 4+7—Od =2r4.2xOd 
&c. &e. &e. . 
WuererorE the fum of the fquares of perpendiculars from 
P, Q to lines touching the circle in the points A, B, C, &c. is 
—anx rex Oc +Of +Od'4&c. But the points c, d, f, 
are in the circumference of a circle, of which the diameter is OQ_ 
or 7, and by Cor. 3. the fum of Oc +Of +Od 4 &c. = OQ x 
into the fum of perpendiculars drawn from O to lines touch- 
ing the circle, of which OQ is the diameter, in the points ¢, d, f, 
&c. Call the fum of thefe perpendiculars s. Then we have 
the fum of the fquares of perpendiculars drawn from P, Q to 
lines touching the circle APQ in the points A, B,C, &c, = 2 1.7% 
ors = (Cor. 3.) AP 4 BP 4OP + &e. + AQ'+BO.+6Q 4&e, 
de 
When the circumference is divided into equal parts by the 
points A, B, C, &c. or the angles at O are equal, s = - x 0Q_ 
n i 
or 5 x7 and 2nT Bars one. 
Ir a regular figure be infcribed in the circle, having its 
angles at the points A, B, C, &c. or a regular figure be cir- 
cumfcribed about the circle, having its fides tangents to it 
in the points A, B, C, &c. we get from the general expreffion 
—4 —4 —-+ ——~}j —— 4 —_4 
Ap + BP +CP + &c. 4, AQ ans 2a 
r 
+4rsaqurpanrs 6 nr, or third proportionals to: ra- 
dius, the chords drawn from either P or Q to the points A, B, 
C, &c. and the cubes of thefe. chords equal, when taken toge- 
ther, to fix times a multiple of the cube of radius by the num- 
ber 
